
Physica D: Nonlinear Phenomena [SJR: 1.049] [HI: 102] [3 followers] Follow Hybrid journal (It can contain Open Access articles) ISSN (Print) 01672789 Published by Elsevier [3042 journals] 
 Wave turbulence theory of elastic plates
 Authors: Gustavo Düring; Christophe Josserand; Sergio Rica
Pages: 42  73
Abstract: Publication date: 15 May 2017
Source:Physica D: Nonlinear Phenomena, Volume 347
Author(s): Gustavo Düring, Christophe Josserand, Sergio Rica
This article presents the complete study of the longtime evolution of random waves of a vibrating thin elastic plate in the limit of small plate deformation so that modes of oscillations interact weakly. According to the wave turbulence theory a nonlinear wave system evolves in longtime creating a slow redistribution of the spectral energy from one mode to another. We derive step by step, following the method of cumulants expansion and multiscale asymptotic perturbations, the kinetic equation for the second order cumulants as well as the second and fourth order renormalization of the dispersion relation of the waves. We characterize the nonequilibrium evolution to an equilibrium wave spectrum, which happens to be the well known Rayleigh–Jeans distribution. Moreover we show the existence of an energy cascade, often called the Kolmogorov–Zakharov spectrum, which happens to be not simply a power law, but a logarithmic correction to the Rayleigh–Jeans distribution. We perform numerical simulations confirming these scenarii, namely the equilibrium relaxation for closed systems and the existence of an energy cascade wave spectrum. Both show a good agreement between theoretical predictions and numerics. We show also some other relevant features of vibrating elastic plates, such as the existence of a selfsimilar wave action inverse cascade which happens to blowup in finite time. We discuss the mechanism of the wave breakdown phenomena in elastic plates as well as the limit of strong turbulence which arises as the thickness of the plate vanishes. Finally, we discuss the role of dissipation and the connection with experiments, and the generalization of the wave turbulence theory to elastic shells.
PubDate: 20170328T02:42:26Z
DOI: 10.1016/j.physd.2017.01.002
Issue No: Vol. 347 (2017)
 Authors: Gustavo Düring; Christophe Josserand; Sergio Rica
 A numerical estimate of the regularity of a family of Strange NonChaotic
Attractors Authors: Lluís Alsedà i Soler; Josep Maria Mondelo González; David Romero i Sànchez
Pages: 74  89
Abstract: Publication date: 15 May 2017
Source:Physica D: Nonlinear Phenomena, Volume 347
Author(s): Lluís Alsedà i Soler, Josep Maria Mondelo González, David Romero i Sànchez
We estimate numerically the regularities of a family of Strange NonChaotic Attractors related with one of the models studied in (Grebogi et al., 1984) (see also Keller, 1996). To estimate these regularities we use wavelet analysis in the spirit of de la Llave and Petrov (2002) together with some adhoc techniques that we develop to overcome the theoretical difficulties that arise in the application of the method to the particular family that we consider. These difficulties are mainly due to the facts that we do not have an explicit formula for the attractor and it is discontinuous almost everywhere for some values of the parameters. Concretely we propose an algorithm based on the Fast Wavelet Transform. Also a quality check of the wavelet coefficients and regularity estimates is done.
PubDate: 20170328T02:42:26Z
DOI: 10.1016/j.physd.2016.12.006
Issue No: Vol. 347 (2017)
 Authors: Lluís Alsedà i Soler; Josep Maria Mondelo González; David Romero i Sànchez
 Full analysis of small hypercycles with shortcircuits in prebiotic
evolution Authors: Josep Sardanyés; J. Tomás Lázaro; Antoni Guillamon; Ernest Fontich
Pages: 90  108
Abstract: Publication date: 15 May 2017
Source:Physica D: Nonlinear Phenomena, Volume 347
Author(s): Josep Sardanyés, J. Tomás Lázaro, Antoni Guillamon, Ernest Fontich
It is known that hypercycles are sensitive to the socalled parasites and shortcircuits. While the impact of parasites has been widely investigated for wellmixed and spatial hypercycles, the effect of shortcircuits in hypercycles remains poorly understood. In this article we analyze the mean field and spatial dynamics of two small, asymmetric hypercycles with shortcircuits. Specifically, we analyze a twomember hypercycle where one of the species contains an autocatalytic loop, as the simplest hypercycle with a shortcircuit. Then, we extend this system by adding another species that closes a threemember hypercycle while keeping the autocatalytic shortcircuit and the twomember cycle. The mean field model allows us to discard the presence of stable or unstable periodic orbits for both systems. We characterize the bifurcations and transitions involved in the dominance of the shortcircuits i.e., in the reduction of the hypercycles’ size. The spatial simulations reveal a randomlike and mixed distribution of the replicators in the allspecies coexistence, ruling out the presence of largescale spatial patterns such as spirals or spots typical of larger, oscillating hypercycles. A Monte Carlo sampling of the parameter space for the wellmixed and the spatial models reveals that the probability of finding stable hypercycles with shortcircuits drastically diminishes from the twomember to the threemember system, especially at growing degradation rates of the replicators. These findings pose a big constraint in the increase of hypercycle’s size and complexity under the presence of inner cycles, suggesting the importance of a rapid growth of hypercycles able to generate spatial structures (e.g., rotating spirals) prior to the emergence of inner cycles. Our results can also be useful for the future design and implementation of synthetic cooperative systems containing catalytic shortcircuits.
PubDate: 20170328T02:42:26Z
DOI: 10.1016/j.physd.2016.12.004
Issue No: Vol. 347 (2017)
 Authors: Josep Sardanyés; J. Tomás Lázaro; Antoni Guillamon; Ernest Fontich
 A nonperturbative analytic expression of signal amplification factor in
stochastic resonance Authors: Asish Kumar Dhara
Pages: 1  44
Abstract: Publication date: 1 April 2017
Source:Physica D: Nonlinear Phenomena, Volume 344
Author(s): Asish Kumar Dhara
We put forward a nonperturbative scheme to calculate the response of an overdamped bistable system driven by a Gaussian white noise and perturbed by a weak monochromatic force (signal) analytically. The formalism takes into account infinite number of perturbation terms of a perturbation series with amplitude of the signal as an expansion parameter. The contributions of infinite number of relaxation modes of the stochastic dynamics to the response are also taken into account in this formalism. A closed form analytic expression of the response is obtained. Only the knowledge of the first nontrivial eigenvalue and the lowest eigenfunction of the unperturbed Fokker–Planck operator are needed to evaluate the response. The response calculated from the derived analytic expression matches fairly well with the numerical results.
PubDate: 20170301T05:09:55Z
DOI: 10.1016/j.physd.2016.11.002
Issue No: Vol. 344 (2017)
 Authors: Asish Kumar Dhara
 Elastic and inelastic collisions of swarms
 Authors: Dieter Armbruster; Stephan Martin; Andrea Thatcher
Pages: 45  57
Abstract: Publication date: 1 April 2017
Source:Physica D: Nonlinear Phenomena, Volume 344
Author(s): Dieter Armbruster, Stephan Martin, Andrea Thatcher
Scattering interactions of swarms in potentials that are generated by an attraction–repulsion model are studied. In free space, swarms in this model form a welldefined steady state describing the translation of a stable formation of the particles whose shape depends on the interaction potential. Thus, the collision between a swarm and a boundary or between two swarms can be treated as (quasi)particle scattering. Such scattering experiments result in internal excitations of the swarm or in bound states, respectively. In addition, varying a parameter linked to the relative importance of damping and potential forces drives transitions between elastic and inelastic scattering of the particles. By tracking the swarm’s center of mass, a refraction rule is derived via simulations relating the incoming and outgoing directions of a swarm hitting the wall. Iterating the map derived from the refraction law allows us to predict and understand the dynamics and bifurcations of swarms in square boxes and in channels.
PubDate: 20170301T05:09:55Z
DOI: 10.1016/j.physd.2016.11.008
Issue No: Vol. 344 (2017)
 Authors: Dieter Armbruster; Stephan Martin; Andrea Thatcher
 Swarming in bounded domains
 Authors: Dieter Armbruster; Sébastien Motsch; Andrea Thatcher
Pages: 58  67
Abstract: Publication date: 1 April 2017
Source:Physica D: Nonlinear Phenomena, Volume 344
Author(s): Dieter Armbruster, Sébastien Motsch, Andrea Thatcher
The Vicsek model is a prototype for the emergence of collective motion. In free space, it is characterized by a swarm of particles all moving in the same direction. Since this dynamic does not include attraction among particles, the swarm, while aligning in velocity space, has no spatial coherence. Adding specular reflection at the boundaries generates global spatial coherence of the swarms while maintaining its velocity alignment. We investigate numerically how the geometry of the domain influences the Vicsek model using three type of geometry: a channel, a disk and a rectangle. Varying the parameters of the Vicsek model (e.g. noise levels and influence horizons), we discuss the mechanisms that generate spatial coherence and show how they create new dynamical solutions of the swarming motions in these geometries. Several observables are introduced to characterize the simulated patterns (e.g. mass profile, center of mass, connectivity of the swarm).
PubDate: 20170301T05:09:55Z
DOI: 10.1016/j.physd.2016.11.009
Issue No: Vol. 344 (2017)
 Authors: Dieter Armbruster; Sébastien Motsch; Andrea Thatcher
 Selfassembly of a filament by curvatureinducing proteins
 Authors: James Kwiecinski; S. Jonathan Chapman; Alain Goriely
Pages: 68  80
Abstract: Publication date: 1 April 2017
Source:Physica D: Nonlinear Phenomena, Volume 344
Author(s): James Kwiecinski, S. Jonathan Chapman, Alain Goriely
We explore a simplified macroscopic model of membrane shaping by means of curvaturesensing BAR proteins. Equations describing the interplay between the shape of a freely floating filament in a fluid and the adhesion kinetics of proteins are derived from mechanical principles. The constant curvature solutions that arise from this system are studied using weakly nonlinear analysis. We show that the stability of the filament’s shape is completely characterized by the parameters associated with protein recruitment and establish that in the bistable regime, proteins aggregate on the filament forming regions of high and low curvatures. This pattern formation is then followed by phasecoarsening that resolves on a timescale dependent on protein diffusion and drift across the filament, which contend to smooth and maintain the pattern respectively. The model is generalized for multiple species of BAR proteins and we show that the stability of the assembled shape is determined by a competition between proteins attaching on opposing sides.
PubDate: 20170301T05:09:55Z
DOI: 10.1016/j.physd.2016.12.001
Issue No: Vol. 344 (2017)
 Authors: James Kwiecinski; S. Jonathan Chapman; Alain Goriely
 Overhanging of membranes and filaments adhering to periodic graph
substrates Authors: Tatsuya Miura
Abstract: Publication date: Available online 20 June 2017
Source:Physica D: Nonlinear Phenomena
Author(s): Tatsuya Miura
This paper mathematically studies membranes and filaments adhering to periodic patterned substrates in a onedimensional model. The problem is formulated by the minimizing problem of an elastic energy with a contact potential on graph substrates. Global minimizers (ground states) are mainly considered in view of their graph representations. Our main results exhibit sufficient conditions for the graph representation and examples of situations where any global minimizer must overhang.
PubDate: 20170622T11:05:22Z
 Authors: Tatsuya Miura
 Distributed synaptic weights in a LIF neural network and learning rules
 Authors: Benoî t Perthame; Delphine Salort; Gilles Wainrib
Abstract: Publication date: Available online 15 June 2017
Source:Physica D: Nonlinear Phenomena
Author(s): Benoî t Perthame, Delphine Salort, Gilles Wainrib
Leaky integrateandfire (LIF) models are meanfield limits, with a large number of neurons, used to describe neural networks. We consider inhomogeneous networks structured by a connectivity parameter (strengths of the synaptic weights) with the effect of processing the input current with different intensities. We first study the properties of the network activity depending on the distribution of synaptic weights and in particular its discrimination capacity. Then, we consider simple learning rules and determine the synaptic weight distribution it generates. We outline the role of noise as a selection principle and the capacity to memorized a learned signal.
PubDate: 20170616T10:48:06Z
DOI: 10.1016/j.physd.2017.05.005
 Authors: Benoî t Perthame; Delphine Salort; Gilles Wainrib
 Multimodel crosspollination in time
 Authors: Hailiang Du; Leonard A. Smith
Abstract: Publication date: Available online 13 June 2017
Source:Physica D: Nonlinear Phenomena
Author(s): Hailiang Du, Leonard A. Smith
The predictive skill of complex models is rarely uniform in modelstate space; in weather forecasting models, for example, the skill of the model can be greater in the regions of most interest to a particular operational agency than it is in “remote” regions of the globe. Given a collection of models, a multimodel forecast system using the crosspollination in time approach can be generalized to take advantage of instances where some models produce forecasts with more information regarding specific components of the modelstate than other models, systematically. This generalization is stated and then successfully demonstrated in a moderate ( ∼ 40 ) dimensional nonlinear dynamical system, suggested by Lorenz, using four imperfect models with similar global forecast skill. Applications to weather forecasting and in economic forecasting are discussed. Given that the relative importance of different phenomena in shaping the weather changes in latitude, changes in attitude among forecast centers in terms of the resources assigned to each phenomena are to be expected. The demonstration establishes that crosspollinating elements of forecast trajectories enriches the collection of simulations upon which the forecast is built, and given the same collection of models can yield a new forecast system with significantly more skill than the original forecast system.
PubDate: 20170616T10:48:06Z
DOI: 10.1016/j.physd.2017.06.001
 Authors: Hailiang Du; Leonard A. Smith
 Wellposedness and dynamics of a fractional stochastic
integrodifferential equation Authors: Linfang Liu; Tomás Caraballo
Abstract: Publication date: Available online 9 June 2017
Source:Physica D: Nonlinear Phenomena
Author(s): Linfang Liu, Tomás Caraballo
In this paper we investigate the wellposedness and dynamics of a fractional stochastic integrodifferential equation describing a reaction process depending on the temperature itself. Existence and uniqueness of solutions of the integrodifferential equation is proved by the Lumer–Phillips theorem. Besides, under appropriate assumptions on the memory kernel and on the magnitude of the nonlinearity, the existence of random attractor is achieved by obtaining first some a priori estimates. Moreover, the random attractor is shown to have finite Hausdorff dimension.
PubDate: 20170612T10:36:04Z
DOI: 10.1016/j.physd.2017.05.006
 Authors: Linfang Liu; Tomás Caraballo
 Analysis of mixedmode oscillationincrementing bifurcations generated in
a nonautonomous constrained Bonhoeffer–van der Pol oscillator Authors: Takuji Kousaka; Yutsuki Ogura; Kuniyasu Shimizu; Hiroyuki Asahara; Naohiko Inaba
Abstract: Publication date: Available online 1 June 2017
Source:Physica D: Nonlinear Phenomena
Author(s): Takuji Kousaka, Yutsuki Ogura, Kuniyasu Shimizu, Hiroyuki Asahara, Naohiko Inaba
Mixedmode oscillations (MMOs) are phenomena observed in a number of dynamic settings, including electrical circuits and chemical systems. Mixedmode oscillationincrementing bifurcations (MMOIBs) are among the most complex MMO bifurcations observed in the large group of MMOgenerating dynamics; however, only a few theoretical analyses of the mechanism causing MMOIBs have been performed to date. In this study, we use a degenerate technique to analyze MMOIBs generated in a Bonhoeffervan der Pol oscillator with a diode under weak periodic perturbation. We consider the idealized case in which the diode operates as an ideal switch; in this case, the governing equation of the oscillator is a piecewise smooth constraint equation and the Poincaré return map is onedimensional, and we find that MMOIBs occur in a manner similar to periodadding bifurcations generated by the circle map. Our numerical results suggest that the universal constant converges to 1.0 and our experimental results demonstrate that MMOIBs can occur successively many times. Our onedimensional Poincaré return map clearly answers the fundamental question of why MMOs are related to Farey sequences even though each MMOgenerating region in the parameter space is terminated by chaos.
PubDate: 20170602T09:28:10Z
DOI: 10.1016/j.physd.2017.05.001
 Authors: Takuji Kousaka; Yutsuki Ogura; Kuniyasu Shimizu; Hiroyuki Asahara; Naohiko Inaba
 Averaging theory at any order for computing limit cycles of discontinuous
piecewise differential systems with many zones Authors: Jaume Llibre; Douglas D. Novaes; Camila A.B. Rodrigues
Abstract: Publication date: Available online 24 May 2017
Source:Physica D: Nonlinear Phenomena
Author(s): Jaume Llibre, Douglas D. Novaes, Camila A.B. Rodrigues
This work is devoted to study the existence of periodic solutions for a class ε family of discontinuous differential systems with many zones. We show that the averaged functions at any order control the existence of crossing limit cycles for systems in this class. We also provide some examples dealing with nonsmooth perturbations of nonlinear centers.
PubDate: 20170528T09:23:39Z
DOI: 10.1016/j.physd.2017.05.003
 Authors: Jaume Llibre; Douglas D. Novaes; Camila A.B. Rodrigues
 The influence of canalization on the robustness of Boolean networks
 Authors: C. Kadelka; J. Kuipers; R. Laubenbacher
Abstract: Publication date: Available online 17 May 2017
Source:Physica D: Nonlinear Phenomena
Author(s): C. Kadelka, J. Kuipers, R. Laubenbacher
Time and statediscrete dynamical systems are frequently used to model molecular networks. This paper provides a collection of mathematical and computational tools for the study of robustness in Boolean network models. The focus is on networks governed by k canalizing functions, a recently introduced class of Boolean functions that contains the wellstudied class of nested canalizing functions. The variable activities and sensitivity of a function quantify the impact of input changes on the function output. This paper generalizes the latter concept to c sensitivity and provides formulas for the activities and c sensitivity of general k canalizing functions as well as canalizing functions with more precisely defined structure. A popular measure for the robustness of a network, the Derrida value, can be expressed as a weighted sum of the c sensitivities of the governing canalizing functions, and can also be calculated for a stochastic extension of Boolean networks. These findings provide a computationally efficient way to obtain Derrida values of Boolean networks, deterministic or stochastic, that does not involve simulation.
PubDate: 20170523T09:10:50Z
DOI: 10.1016/j.physd.2017.05.002
 Authors: C. Kadelka; J. Kuipers; R. Laubenbacher
 Optimal strategies for the control of autonomous vehicles in data
assimilation Authors: D. McDougall; R.O. Moore
Abstract: Publication date: Available online 5 May 2017
Source:Physica D: Nonlinear Phenomena
Author(s): D. McDougall, R.O. Moore
We propose a method to compute optimal control paths for autonomous vehicles deployed for the purpose of inferring a velocity field. In addition to being advected by the flow, the vehicles are able to effect a fixed relative speed with arbitrary control over direction. It is this direction that is used as the basis for the locally optimal control algorithm presented here, with objective formed from the variance trace of the expected posterior distribution. We present results for linear flows near hyperbolic fixed points.
PubDate: 20170507T15:23:30Z
DOI: 10.1016/j.physd.2017.04.001
 Authors: D. McDougall; R.O. Moore
 Solution landscapes in nematic microfluidics
 Authors: M. Crespo; A. Majumdar; A.M. Ramos; I.M. Griffiths
Abstract: Publication date: Available online 4 May 2017
Source:Physica D: Nonlinear Phenomena
Author(s): M. Crespo, A. Majumdar, A.M. Ramos, I.M. Griffiths
We study the static equilibria of a simplified Leslie–Ericksen model for a unidirectional uniaxial nematic flow in a prototype microfluidic channel, as a function of the pressure gradient G and inverse anchoring strength, B . We numerically find multiple static equilibria for admissible pairs ( G , B ) and classify them according to their winding numbers and stability. The case G = 0 is analytically tractable and we numerically study how the solution landscape is transformed as G increases. We study the onedimensional dynamical model, the sensitivity of the dynamic solutions to initial conditions and the rate of change of G and B . We provide a physically interesting example of how the time delay between the applications of G and B can determine the selection of the final steady state.
PubDate: 20170507T15:23:30Z
DOI: 10.1016/j.physd.2017.04.004
 Authors: M. Crespo; A. Majumdar; A.M. Ramos; I.M. Griffiths
 Invariant manifolds and the parameterization method in coupled energy
harvesting piezoelectric oscillators Authors: Albert Granados
Abstract: Publication date: Available online 19 April 2017
Source:Physica D: Nonlinear Phenomena
Author(s): Albert Granados
Energy harvesting systems based on oscillators aim to capture energy from mechanical oscillations and convert it into electrical energy. Widely extended are those based on piezoelectric materials, whose dynamics are Hamiltonian submitted to different sources of dissipation: damping and coupling. These dissipations bring the system to low energy regimes, which is not desired in long term as it diminishes the absorbed energy. To avoid or to minimize such situations, we propose that the coupling of two oscillators could benefit from theory of Arnold diffusion. Such phenomenon studies O ( 1 ) energy variations in Hamiltonian systems and hence could be very useful in energy harvesting applications. This article is a first step towards this goal. We consider two piezoelectric beams submitted to a small forcing and coupled through an electric circuit. By considering the coupling, damping and forcing as perturbations, we prove that the unperturbed system possesses a 4dimensional Normally Hyperbolic Invariant Manifold with 5 and 4dimensional stable and unstable manifolds, respectively. These are locally unique after the perturbation. By means of the parameterization method, we numerically compute parameterizations of the perturbed manifold, its stable and unstable manifolds and study its inner dynamics. We show evidence of homoclinic connections when the perturbation is switched on.
PubDate: 20170425T10:34:15Z
DOI: 10.1016/j.physd.2017.04.003
 Authors: Albert Granados
 The spectrum of the torus profile to a geometric variational problem with
long range interaction Authors: Xiaofeng Ren; Juncheng Wei
Abstract: Publication date: Available online 18 April 2017
Source:Physica D: Nonlinear Phenomena
Author(s): Xiaofeng Ren, Juncheng Wei
The profile problem for the OhtaKawasaki diblock copolymer theory is a geometric variational problem. The energy functional is defined on sets in R 3 of prescribed volume and the energy of an admissible set is its perimeter plus a long range interaction term related to the Newtonian potential of the set. This problem admits a solution, called a torus profile, that is a set enclosed by an approximate torus of the major radius 1 and the minor radius q . The torus profile is both axially symmetric about the z axis and reflexively symmetric about the x y plane. There is a way to set up the profile problem in a function space as a partial differentialintegro equation. The linearized operator L of the problem at the torus profile is decomposed into a family of linear ordinary differentialintegro operators L m where the index m = 0 , 1 , 2 , . . . is called a mode. The spectrum of L is the union of the spectra of the L m ’s. It is proved that for each m , when q is sufficiently small, L m is positive definite. ( 0 is an eigenvalue for both L 0 and L 1 , due to the translation and rotation invariance.) As q tends to 0 , more and more L m ’s become positive definite. However no matter how small q is, there is always a mode m of which L m has a negative eigenvalue. This mode grows to infinity like q − 3 / 4 as q → 0 .
PubDate: 20170425T10:34:15Z
DOI: 10.1016/j.physd.2017.01.001
 Authors: Xiaofeng Ren; Juncheng Wei
 Dynamical and energetic instabilities of F=2 spinor BoseEinstein
condensates in an optical lattice Authors: DengShan Wang; YuRen Shi; WenXing Feng; Lin Wen
Abstract: Publication date: Available online 17 April 2017
Source:Physica D: Nonlinear Phenomena
Author(s): DengShan Wang, YuRen Shi, WenXing Feng, Lin Wen
The dynamical and energetic instabilities of the F =2 spinor BoseEinstein condensates in an optical lattice are investigated theoretically and numerically. By analyzing the dynamical response of different carrier waves to an additional linear perturbation, we obtain the instability criteria for the ferromagnetic, uniaxial nematic, biaxial nematic and cyclic states, respectively. When an external magnetic field is taken into account, we find that the linear or quadratic Zeeman effects obviously affect the dynamical instability properties of uniaxial nematic, biaxial nematic and cyclic states, but not for the ferromagnetic one. In particular, it is found that the faster moving F =2 spinor BEC has a larger energetic instability region than lower one in all the four states. In addition, it is seen that for most states there probably exists a critical value k c > 0 , for which k > k c gives the energetic instability to arise under appreciative parameters.
PubDate: 20170418T03:43:03Z
DOI: 10.1016/j.physd.2017.04.002
 Authors: DengShan Wang; YuRen Shi; WenXing Feng; Lin Wen
 On the phenomenon of mixed dynamics in PikovskyTopaj system of coupled
rotators Authors: A.S. Gonchenko; S.V. Gonchenko; A.O. Kazakov; D.V. Turaev
Abstract: Publication date: Available online 30 March 2017
Source:Physica D: Nonlinear Phenomena
Author(s): A.S. Gonchenko, S.V. Gonchenko, A.O. Kazakov, D.V. Turaev
A oneparameter family of timereversible systems on threedimensional torus is considered. It is shown that the dynamics is not conservative, namely the attractor and repeller intersect but not coincide. We explain this as the manifestation of the socalled mixed dynamics phenomenon which corresponds to a persistent intersection of the closure of the stable periodic orbits and the closure of the completely unstable periodic orbits. We search for the stable and unstable periodic orbits indirectly, by finding nonconservative saddle periodic orbits and heteroclinic connections between them. In this way, we are able to claim the existence of mixed dynamics for a large range of parameter values. We investigate local and global bifurcations that can be used for the detection of mixed dynamics.
PubDate: 20170404T03:13:49Z
DOI: 10.1016/j.physd.2017.02.002
 Authors: A.S. Gonchenko; S.V. Gonchenko; A.O. Kazakov; D.V. Turaev
 Pattern formation on the free surface of a ferrofluid: Spatial dynamics
and homoclinic bifurcation Authors: M.D. Groves; D.J.B. Lloyd; A. Stylianou
Abstract: Publication date: Available online 23 March 2017
Source:Physica D: Nonlinear Phenomena
Author(s): M.D. Groves, D.J.B. Lloyd, A. Stylianou
We establish the existence of spatially localised onedimensional free surfaces of a ferrofluid near onset of the Rosensweig instability, assuming a general (nonlinear) magnetisation law. It is shown that the ferrohydrostatic equations can be derived from a variational principle that allows one to formulate them as an (infinitedimensional) spatial Hamiltonian system in which the unbounded freesurface direction plays the role of time. A centremanifold reduction technique converts the problem for small solutions near onset to an equivalent Hamiltonian system with finitely many degrees of freedom. Normalform theory yields the existence of homoclinic solutions to the reduced system, which correspond to spatially localised solutions of the ferrohydrostatic equations.
PubDate: 20170328T02:42:26Z
DOI: 10.1016/j.physd.2017.03.004
 Authors: M.D. Groves; D.J.B. Lloyd; A. Stylianou
 Synchronization of weakly coupled canard oscillators
 Authors: Elif Köksal Ersöz; Mathieu Desroches; Martin Krupa
Abstract: Publication date: Available online 22 March 2017
Source:Physica D: Nonlinear Phenomena
Author(s): Elif Köksal Ersöz, Mathieu Desroches, Martin Krupa
Synchronization has been studied extensively in the context of weakly coupled oscillators using the socalled phase response curve (PRC) which measures how a change of the phase of an oscillator is affected by a small perturbation. This approach was based upon the work of Malkin, and it has been extended to relaxation oscillators. Namely, synchronization conditions were established under the weak coupling assumption, leading to a criterion for the existence of synchronous solutions of weakly coupled relaxation oscillators. Previous analysis relies on the fact that the slow nullcline does not intersect the fast nullcline near one of its fold points, where canard solutions can arise. In the present study we use numerical continuation techniques to solve the adjoint equations and we show that synchronization properties of canard cycles are different than those of classical relaxation cycles. In particular, we highlight a new special role of the maximal canard in separating two distinct synchronization regimes: the Hopf regime and the relaxation regime. Phase plane analysis of slowfast oscillators undergoing a canard explosion provides an explanation for this change of synchronization properties across the maximal canard.
PubDate: 20170328T02:42:26Z
DOI: 10.1016/j.physd.2017.02.016
 Authors: Elif Köksal Ersöz; Mathieu Desroches; Martin Krupa
 Integrable systems and invariant curve flows in symplectic Grassmannian
space Authors: Junfeng Song; Changzheng Qu; Ruoxia Yao
Abstract: Publication date: Available online 21 March 2017
Source:Physica D: Nonlinear Phenomena
Author(s): Junfeng Song, Changzheng Qu, Ruoxia Yao
In this paper, local geometry of curves in the symplectic Grassmannian homogeneous space Sp ( 4 , R ) / ( Sp ( 2 , R ) × Sp ( 2 , R ) ) and its connection with that of the pseudohyperbolic space H 2 , 2 are studied. The groupbased SerretFrenet equations and the associated MaurerCartan differential invariants for the Grassmannian curves are obtained by using the equivariant moving frame method. The Grassmannian natural frame are also constructed by a gauge transformation from the SerretFrenet frame, relating to the hyperbolic natural frame by the local Lie group isomorphism. Using the natural frames, invariant curve flows in the Grassmannian and the hyperbolic spaces are studied. It is shown that certain intrinsic curve flows induce the biHamiltonian integrable matrix mKdV equation on the MaurerCartan differential invariants.
PubDate: 20170328T02:42:26Z
DOI: 10.1016/j.physd.2017.02.013
 Authors: Junfeng Song; Changzheng Qu; Ruoxia Yao
 Stability of the phase motion in racetrack microtons
 Authors: Yu.A. Kubyshin; O. Larreal; R. RamírezRos; T.M. Seara
Abstract: Publication date: Available online 18 March 2017
Source:Physica D: Nonlinear Phenomena
Author(s): Yu.A. Kubyshin, O. Larreal, R. RamírezRos, T.M. Seara
We model the phase oscillations of electrons in racetrack microtrons by means of an area preserving map with a fixed point at the origin, which represents the synchronous trajectory of a reference particle in the beam. We study the nonlinear stability of the origin in terms of the synchronous phase —the phase of the synchronous particle at the injection. We estimate the size and shape of the stability domain around the origin, whose main connected component is enclosed by an invariant curve. We describe the evolution of the stability domain as the synchronous phase varies. We also clarify the role of the stable and unstable invariant curves of some hyperbolic (fixed or periodic) points.
PubDate: 20170320T19:31:07Z
DOI: 10.1016/j.physd.2017.03.001
 Authors: Yu.A. Kubyshin; O. Larreal; R. RamírezRos; T.M. Seara
 Markovian properties of velocity increments in boundary layer turbulence
 Authors: Murat Tutkun
Abstract: Publication date: Available online 16 March 2017
Source:Physica D: Nonlinear Phenomena
Author(s): Murat Tutkun
Markovian properties of the turbulent velocity increments in a flat plate boundary layer at Re θ = 19100 are investigated using hotwire anemometry measurements of the streamwise velocity component in a wind tunnel. Increments of the longitudinal velocities at different wallnormal positions show that the flow exhibits Markovian properties when the separation between different scales, or the MarkovEinstein coherence length, is on the order of the Taylor microscale, λ . The results indicate that Markovian nature of turbulence evolves across the boundary layer showing certain characteristics pertaining to the distance to the wall. The connection between the Markovian properties of turbulent boundary layer and existence of the spectral gap is explored. Markovianity of the process is also discussed in relation to the nonlocal nonlinear versus local nonlinear transfer of energy, triadic interactions and dissipation.
PubDate: 20170320T19:31:07Z
DOI: 10.1016/j.physd.2017.03.002
 Authors: Murat Tutkun
 Targeted energy transfer in laminar vortexinduced vibration of a sprung
cylinder with a nonlinear dissipative rotator Authors: Antoine Blanchard; Lawrence A. Bergman; Alexander F. Vakakis
Abstract: Publication date: Available online 16 March 2017
Source:Physica D: Nonlinear Phenomena
Author(s): Antoine Blanchard, Lawrence A. Bergman, Alexander F. Vakakis
We computationally investigate the dynamics of a linearlysprung circular cylinder immersed in an incompressible flow and undergoing transverse vortexinduced vibration (VIV), to which is attached a rotational nonlinear energy sink (NES) consisting of a mass that freely rotates at constant radius about the cylinder axis, and whose motion is restrained by a rotational linear viscous damper. The inertial coupling between the rotational motion of the attached mass and the rectilinear motion of the cylinder is “essentially nonlinear”, which, in conjunction with dissipation, allows for oneway, nearly irreversible targeted energy transfer (TET) from the oscillating cylinder to the nonlinear dissipative attachment. At the intermediate Reynolds number R e = 100 , the NESequipped sprung cylinder undergoes repetitive cycles of slowly decaying oscillations punctuated by intervals of chaotic instabilities. During the slowly decaying portion of each cycle, the dynamics of the cylinder is regular and, for large enough values of the ratio ε of the NES mass to the total mass (i.e., NES mass plus cylinder mass), can lead to significant vortex street elongation with partial stabilization of the wake. As ε approaches zero, no such vortex elongation is observed and the wake patterns appear similar to that for a sprung cylinder with no NES. We apply proper orthogonal decomposition (POD) to the velocity flow field during a slowly decaying portion of the solution and show that, in situations where vortex elongation occurs, the NES, though not in direct contact with the surrounding fluid, has a drastic effect on the underlying flow structures, imparting significant and continuous passive redistribution of energy among POD modes. We construct a PODbased reducedorder model for the lift coefficient to characterize energy transactions between the fluid and the cylinder throughout the slowly decaying cycle. We introduce a quantitative signed measure of the work done by the fluid on the cylinder over one quasiperiod of the slowly decaying response and find that vortex elongation is associated with a sign change of that measure, indicating that a reversal of the direction of energy transfer, with the cylinder “leaking energy back” to the flow, is responsible for partial stabilization and elongation of the wake. We interpret these findings in terms of the spatial structure and energy distribution of the POD modes, and relate them to the mechanism of transient resonance capture into a slow invariant manifold of the fluid–structure interaction dynamics.
PubDate: 20170320T19:31:07Z
DOI: 10.1016/j.physd.2017.03.003
 Authors: Antoine Blanchard; Lawrence A. Bergman; Alexander F. Vakakis
 Fluctuations, response, and resonances in a simple atmospheric model
 Authors: Andrey Gritsun; Valerio Lucarini
Abstract: Publication date: Available online 14 March 2017
Source:Physica D: Nonlinear Phenomena
Author(s): Andrey Gritsun, Valerio Lucarini
We study the response of a simple quasigeostrophic barotropic model of the atmosphere to various classes of perturbations affecting its forcing and its dissipation using the formalism of the Ruelle response theory. We investigate the geometry of such perturbations by constructing the covariant Lyapunov vectors of the unperturbed system and discover in one specific case–orographic forcing–a substantial projection of the forcing onto the stable directions of the flow. This results into a resonant response shaped as a Rossbylike wave that has no resemblance to the unforced variability in the same range of spatial and temporal scales. Such a climatic surprise corresponds to a violation of the fluctuationdissipation theorem, in agreement with the basic tenets of nonequilibrium statistical mechanics. The resonance can be attributed to a specific group of rarely visited unstable periodic orbits of the unperturbed system. Our results reinforce the idea of using basic methods of nonequilibrium statistical mechanics and highdimensional chaotic dynamical systems to approach the problem of understanding climate dynamics.
PubDate: 20170320T19:31:07Z
DOI: 10.1016/j.physd.2017.02.015
 Authors: Andrey Gritsun; Valerio Lucarini
 Standing and travelling waves in a spherical brain model: The Nunez model
revisited Authors: S. Visser; R. Nicks; O. Faugeras; S. Coombes
Abstract: Publication date: Available online 8 March 2017
Source:Physica D: Nonlinear Phenomena
Author(s): S. Visser, R. Nicks, O. Faugeras, S. Coombes
The Nunez model for the generation of electroencephalogram (EEG) signals is naturally described as a neural field model on a sphere with spacedependent delays. For simplicity, dynamical realisations of this model either as a damped wave equation or an integrodifferential equation, have typically been studied in idealised one dimensional or planar settings. Here we revisit the original Nunez model to specifically address the role of spherical topology on spatiotemporal pattern generation. We do this using a mixture of Turing instability analysis, symmetric bifurcation theory, center manifold reduction and direct simulations with a bespoke numerical scheme. In particular we examine standing and travelling wave solutions using normal form computation of primary and secondary bifurcations from a steady state. Interestingly, we observe spatiotemporal patterns which have counterparts seen in the EEG patterns of both epileptic and schizophrenic brain conditions.
PubDate: 20170312T21:31:41Z
DOI: 10.1016/j.physd.2017.02.017
 Authors: S. Visser; R. Nicks; O. Faugeras; S. Coombes
 Controlling coexisting attractors of an impacting system via linear
augmentation Authors: Yang Liu; Joseph Páez Chávez
Abstract: Publication date: Available online 8 March 2017
Source:Physica D: Nonlinear Phenomena
Author(s): Yang Liu, Joseph Páez Chávez
This paper studies the control of coexisting attractors in an impacting system via a recently developed control law based on linear augmentation. Special attention is given to two control issues in the framework of multistable engineering systems, namely, the switching between coexisting attractors without altering the system’s main parameters and the avoidance of grazinginduced chaotic responses. The effectiveness of the proposed control scheme is confirmed numerically for the case of a periodically excited, soft impact oscillator. Our analysis shows how pathfollowing techniques for nonsmooth systems can be used in order to determine the optimal control parameters in terms of energy expenditure due to the control signal and transient behavior of the control error, which can be applied to a broad range of engineering problems.
PubDate: 20170312T21:31:41Z
DOI: 10.1016/j.physd.2017.02.018
 Authors: Yang Liu; Joseph Páez Chávez
 Construction of Darboux coordinates and PoincaréBirkhoff normal forms in
noncanonical Hamiltonian systems Authors: Andrej Junginger; Jörg Main; Günter Wunner
Abstract: Publication date: Available online 7 March 2017
Source:Physica D: Nonlinear Phenomena
Author(s): Andrej Junginger, Jörg Main, Günter Wunner
We demonstrate a general method to construct Darboux coordinates via normal form expansions in noncanonical Hamiltonian system obtained from e.g. a variational approach to quantum systems. The procedure serves as a tool to naturally extract canonical coordinates out of the variational parameters and at the same time to transform the energy functional into its PoincaréBirkhoff normal form. The method is general in the sense that it is applicable for arbitrary degrees of freedom, in arbitrary orders of the local expansion, and it is independent of the precise form of the Hamilton operator. The method presented allows for the general and systematic investigation of quantum systems in the vicinity of fixed points, which e.g. correspond to ground, excited or transition states. Moreover, it directly allows to calculate classical and quantum reaction rates by applying transition state theory.
PubDate: 20170312T21:31:41Z
DOI: 10.1016/j.physd.2017.02.014
 Authors: Andrej Junginger; Jörg Main; Günter Wunner
 Assessment of the effects of azimuthal mode number perturbations upon the
implosion processes of fluids in cylinders Authors: Michael Lindstrom
Abstract: Publication date: Available online 1 March 2017
Source:Physica D: Nonlinear Phenomena
Author(s): Michael Lindstrom
Fluid instabilities arise in a variety of contexts and are often unwanted results of engineering imperfections. In one particular model for a magnetized target fusion reactor, a pressure wave is propagated in a cylindrical annulus comprised of a dense fluid before impinging upon a plasma and imploding it. Part of the success of the apparatus is a function of how axiallysymmetric the final pressure pulse is upon impacting the plasma. We study a simple model for the implosion of the system to study how imperfections in the pressure imparted on the outer circumference grow due to geometric focusing. Our methodology entails linearizing the compressible Euler equations for mass and momentum conservation about a cylindrically symmetric problem and analyzing the perturbed profiles at different mode numbers. The linearized system gives rise to singular shocks and through analyzing the perturbation profiles at various times, we infer that high mode numbers are dampened through the propagation. We also study the Linear KleinGordon equation in the context of stability of linear cylindrical wave formation whereby highly oscillatory, bounded behaviour is observed in a far field solution.
PubDate: 20170307T21:14:06Z
DOI: 10.1016/j.physd.2017.02.012
 Authors: Michael Lindstrom
 KdV cnoidal waves in a traffic flow model with periodic boundaries
 Authors: L.L. Hattam
Abstract: Publication date: Available online 1 March 2017
Source:Physica D: Nonlinear Phenomena
Author(s): L.L. Hattam
An optimalvelocity (OV) model describes car motion on a single lane road. In particular, near to the boundary signifying the onset of traffic jams, this model reduces to a perturbed Kortewegde Vries (KdV) equation using asymptotic analysis. Previously, the KdV soliton solution has then been found and compared to numerical results (see Muramatsu and Nagatani (1999)). Here, we instead apply modulation theory to this perturbed KdV equation to obtain at leading order, the modulated cnoidal wave solution. At the next order, the Whitham equations are derived, which have been modified due to the equation perturbation terms. Next, from this modulation system, a family of spatially periodic cnoidal waves are identified that characterise vehicle headway distance. Then, for this set of solutions, we establish the relationship between the wave speed, the modulation term and the driver sensitivity. This analysis is confirmed with comparisons to numerical solutions of the OV model. As well, the longtime behaviour of these solutions is investigated.
PubDate: 20170307T21:14:06Z
DOI: 10.1016/j.physd.2017.02.010
 Authors: L.L. Hattam
 Controlling roughening processes in the stochastic KuramotoSivashinsky
equation Authors: S.N. Gomes; S. Kalliadasis; D.T. Papageorgiou; G.A. Pavliotis; M. Pradas
Abstract: Publication date: Available online 1 March 2017
Source:Physica D: Nonlinear Phenomena
Author(s): S.N. Gomes, S. Kalliadasis, D.T. Papageorgiou, G.A. Pavliotis, M. Pradas
We present a novel control methodology to control the roughening processes of semilinear parabolic stochastic partial differential equations in one dimension, which we exemplify with the stochastic KuramotoSivashinsky equation. The original equation is split into a linear stochastic and a nonlinear deterministic equation so that we can apply linear feedback control methods. Our control strategy is then based on two steps: first, stabilize the zero solution of the deterministic part and, second, control the roughness of the stochastic linear equation. We consider both periodic controls and point actuated ones, observing in all cases that the second moment of the solution evolves in time according to a powerlaw until it saturates at the desired controlled value.
PubDate: 20170307T21:14:06Z
DOI: 10.1016/j.physd.2017.02.011
 Authors: S.N. Gomes; S. Kalliadasis; D.T. Papageorgiou; G.A. Pavliotis; M. Pradas
 A novel route to a Hopf bifurcation scenario in switched systems with
deadzone Authors: P. Kowalczyk
Abstract: Publication date: Available online 1 March 2017
Source:Physica D: Nonlinear Phenomena
Author(s): P. Kowalczyk
Planar switched systems with deadzone are analyzed. In particular, we consider the effects of a perturbation which is applied to a linear control law and, due to the perturbation, the control changes from purely positional to positionvelocity control. This type of a perturbation leads to a novel Hopflike discontinuity induced bifurcation. We show that this bifurcation leads to the creation of a small scale limit cycle attractor, which scales as the square root of the bifurcation parameter. We then investigate numerically a planar switched system with a positional feedback law, deadzone and time delay in the switching function. Using the same parameter values as for the switched system without time delay in the switching function, we show a Hopflike bifurcation scenario which exhibits a qualitative and a quantitative agreement with the scenario analyzed for the nondelayed system.
PubDate: 20170307T21:14:06Z
DOI: 10.1016/j.physd.2017.02.007
 Authors: P. Kowalczyk
 Hopf and Homoclinic Bifurcations on the sliding vector field of switching
systems in R3: A case study in power electronics Authors: Rony Cristiano; Tiago Carvalho; Durval J. Tonon; Daniel J. Pagano
Abstract: Publication date: Available online 24 February 2017
Source:Physica D: Nonlinear Phenomena
Author(s): Rony Cristiano, Tiago Carvalho, Durval J. Tonon, Daniel J. Pagano
In this paper, Hopf and homoclinic bifurcations that occur in the sliding vector field of switching systems in R 3 are studied. In particular, a dcdc boost converter with sliding mode control and washout filter is analyzed. This device is modelled as a threedimensional Filippov system, characterized by the existence of sliding movement and restricted to the switching manifold. The operating point of the converter is a stable pseudoequilibrium and it undergoes a subcritical Hopf bifurcation. Such a bifurcation occurs in the sliding vector field and creates, in this field, an unstable limit cycle. The limit cycle is connected to the switching manifold and disappears when it touches the visibleinvisible twofold point, resulting in an homoclinic loop which itself closes in this twofold point. The study of these dynamic phenomena that can be found in different power electronic circuits controlled by sliding mode control strategies are relevant from the viewpoint of the global stability and robustness of the control design.
PubDate: 20170301T05:09:55Z
DOI: 10.1016/j.physd.2017.02.005
 Authors: Rony Cristiano; Tiago Carvalho; Durval J. Tonon; Daniel J. Pagano
 Axisymmetric pulse train solutions in narrowgap spherical Couette flow
 Authors: Adam Child; Rainer Hollerbach; Evy Kersalé
Abstract: Publication date: Available online 24 February 2017
Source:Physica D: Nonlinear Phenomena
Author(s): Adam Child, Rainer Hollerbach, Evy Kersalé
We numerically compute the flow induced in a spherical shell by fixing the outer sphere and rotating the inner one. The aspect ratio ϵ = ( r o − r i ) / r i is set at 0.04 and 0.02, and in each case the Reynolds number measuring the inner sphere’s rotation rate is increased to ∼ 10% beyond the first bifurcation from the basic state flow. For ϵ = 0.04 the initial bifurcations are the same as in previous numerical work at ϵ = 0.154 , and result in steady one and twovortex states. Further bifurcations yield travelling wave solutions similar to previous analytic results valid in the ϵ → 0 limit. For ϵ = 0.02 the steady onevortex state no longer exists, and the first bifurcation is directly to these travelling wave solutions, consisting of pulse trains of Taylor vortices travelling toward the equator from both hemispheres, and annihilating there in distinct phaseslip events. We explore these timedependent solutions in detail, and find that they can be both equatorially symmetric and asymmetric, as well as periodic or quasiperiodic in time.
PubDate: 20170301T05:09:55Z
DOI: 10.1016/j.physd.2017.02.009
 Authors: Adam Child; Rainer Hollerbach; Evy Kersalé
 Assimilating Eulerian and Lagrangian data in trafficflow models
 Authors: Chao Xia; Courtney Cochrane; Joseph DeGuire; Gaoyang Fan; Emma Holmes; Melissa McGuirl; Patrick Murphy; Jenna Palmer; Paul Carter; Laura Slivinski; Björn Sandstede
Abstract: Publication date: Available online 21 February 2017
Source:Physica D: Nonlinear Phenomena
Author(s): Chao Xia, Courtney Cochrane, Joseph DeGuire, Gaoyang Fan, Emma Holmes, Melissa McGuirl, Patrick Murphy, Jenna Palmer, Paul Carter, Laura Slivinski, Björn Sandstede
Data assimilation of traffic flow remains a challenging problem. One difficulty is that data come from different sources ranging from stationary sensors and camera data to GPS and cell phone data from moving cars. Sensors and cameras give information about traffic density, while GPS data provide information about the positions and velocities of individual cars. Previous methods for assimilating Lagrangian data collected from individual cars relied on specific properties of the underlying computational model or its reformulation in Lagrangian coordinates. These approaches make it hard to assimilate both Eulerian density and Lagrangian positional data simultaneously. In this paper, we propose an alternative approach that allows us to assimilate both Eulerian and Lagrangian data. We show that the proposed algorithm is accurate and works well in different traffic scenarios and regardless of whether ensemble Kalman or particle filters are used. We also show that the algorithm is capable of estimating parameters and assimilating real traffic observations and synthetic observations obtained from microscopic models.
PubDate: 20170221T09:49:54Z
DOI: 10.1016/j.physd.2017.02.004
 Authors: Chao Xia; Courtney Cochrane; Joseph DeGuire; Gaoyang Fan; Emma Holmes; Melissa McGuirl; Patrick Murphy; Jenna Palmer; Paul Carter; Laura Slivinski; Björn Sandstede
 Complex predator invasion waves in a HollingTanner model with nonlocal
prey interaction Authors: A. Bayliss; V.A. Volpert
Abstract: Publication date: Available online 20 February 2017
Source:Physica D: Nonlinear Phenomena
Author(s): A. Bayliss, V.A. Volpert
We consider predator invasions for the nonlocal HollingTanner model. Predators are introduced in a small region adjacent to an extensive predatorfree region. In its simplest form an invasion front propagates into the predatorfree region with a predatorprey coexistence state displacing the predatorfree state. However, patterns may form in the wake of the invasion front due to instability of the coexistence state. The coexistence state can be subject to either oscillatory or cellular instability, depending on parameters. Furthermore, the oscillatory instability can be either at zero wave number or finite wave number. In addition, the (unstable) predatorfree state can be subject to additional cellular instabilities when the extent of the nonlocality is sufficiently large. We perform numerical simulations that demonstrate that the invasion wave may have a complex structure in which different spatial regions exhibit qualitatively different behaviors. These regions are separated by relatively narrow transition regions that we refer to as fronts. We also derive analytic approximations for the speeds of the fronts and find qualitative and quantitative agreement with the results of computations.
PubDate: 20170221T09:49:54Z
DOI: 10.1016/j.physd.2017.02.003
 Authors: A. Bayliss; V.A. Volpert
 Elementary solutions for a model Boltzmann equation in one dimension and
the connection to grossly determined solutions Authors: Thomas E. Carty
Abstract: Publication date: Available online 20 February 2017
Source:Physica D: Nonlinear Phenomena
Author(s): Thomas E. Carty
The Fouriertransformed version of the time dependent slipflow model Boltzmann equation associated with the linearized BGK model is solved in order to determine the solution’s asymptotics. The ultimate goal of this paper is to demonstrate that there exists a robust set of solutions to this model Boltzmann equation that possess a special property that was conjectured by Truesdell and Muncaster: that solutions decay to a subclass of the solution set uniquely determined by the initial mass density of the gas called the grossly determined solutions. First we determine the spectrum and eigendistributions of the associated homogeneous equation. Then, using Case’s method of elementary solutions, we find analytic timedependent solutions to the model Boltzmann equation for initial data with a specialized compact support condition under the Fourier transform. In doing so, we show that the spectrum separates the solutions into two distinct parts: one that behaves as a set of transient solutions and the other limiting to a stable subclass of solutions. Thus, we demonstrate that for gas flows with this specialized initial density condition, in time all gas flows for the one dimensional model Boltzmann equation act as grossly determined solutions.
PubDate: 20170221T09:49:54Z
DOI: 10.1016/j.physd.2017.02.008
 Authors: Thomas E. Carty
 Bright and dark solitons in the unidirectional long wave limit for the
energy transfer on anharmonic crystal lattices Authors: Luis A. CisnerosAke; José F. Solano Peláez
Abstract: Publication date: Available online 9 February 2017
Source:Physica D: Nonlinear Phenomena
Author(s): Luis A. CisnerosAke, José F. Solano Peláez
The problem of energy transportation along a cubic anharmonic crystal lattice, in the unidirectional long wave limit, is considered. A detailed process, in the discrete lattice equations, shows that unidirectional stable propagating waves for the continuum limit produces a coupled system between a nonlinear Schrödinger (NLS) equation and the KortewegdeVries (KdV) equation. The traveling wave formalism provides a diversity of exact solutions ranging from the classical Davydov’s soliton (subsonic and supersonic) of the first and second kind to a class consisting in the coupling between the KdV soliton and dark solitons containing the typical ones (similar to the darkgray soliton in the standard defocusing NLS) and a new kind in the form of a twohump dark soliton. This family of exact solutions are numerically tested, by means of the pseudo spectral method, in our NLSKdV system.
PubDate: 20170215T09:34:42Z
DOI: 10.1016/j.physd.2017.02.001
 Authors: Luis A. CisnerosAke; José F. Solano Peláez
 The LyapunovKrasovskii theorem and a sufficient criterion for local
stability of isochronal synchronization in networks of delaycoupled
oscillators Authors: J.M.V. Grzybowski; E.E.N. Macau; T. Yoneyama
Abstract: Publication date: Available online 9 February 2017
Source:Physica D: Nonlinear Phenomena
Author(s): J.M.V. Grzybowski, E.E.N. Macau, T. Yoneyama
This paper presents a selfcontained framework for the stability assessment of isochronal synchronization in networks of chaotic and limitcycle oscillators. The results were based on the LyapunovKrasovskii theorem and they establish a sufficient condition for local synchronization stability of as a function of the system and network parameters. With this in mind, a network of mutually delaycoupled oscillators subject to direct selfcoupling is considered and then the resulting error equations are blockdiagonalized for the purpose of studying their stability. These error equations are evaluated by means of analytical stability results derived from the LyapunovKrasovskii theorem. The proposed approach is shown to be a feasible option for the investigation of local stability of isochronal synchronization for a variety of oscillators coupled through linear functions of the state variables under a given undirected graph structure. This ultimately permits the systematic identification of stability regions within the highdimensionality of the network parameter space. Examples of applications of the results to a number of networks of delaycoupled chaotic and limitcycle oscillators are provided, such as Lorenz, Rössler, Cubic Chua’s circuit, Van der Pol oscillator and the HindmarshRose neuron.
PubDate: 20170215T09:34:42Z
DOI: 10.1016/j.physd.2017.01.005
 Authors: J.M.V. Grzybowski; E.E.N. Macau; T. Yoneyama
 Wave fronts and cascades of soliton interactions in the periodic two
dimensional Volterra system Authors: Rhys Bury; Alexander V. Mikhailov; Jing Ping Wang
Abstract: Publication date: Available online 6 February 2017
Source:Physica D: Nonlinear Phenomena
Author(s): Rhys Bury, Alexander V. Mikhailov, Jing Ping Wang
In the paper we develop the dressing method for the solution of the twodimensional periodic Volterra system with a period N . We derive soliton solutions of arbitrary rank k and give a full classification of rank 1 solutions. We have found a new class of exact solutions corresponding to wave fronts which represent smooth interfaces between two nonlinear periodic waves or a periodic wave and a trivial (zero) solution. The wave fronts are nonstationary and they propagate with a constant average velocity. The system also has soliton solutions similar to breathers, which resembles soliton webs in the KP theory. We associate the classification of soliton solutions with the Schubert decomposition of the Grassmannians Gr R ( k , N ) and Gr C ( k , N ) .
PubDate: 20170209T06:01:16Z
DOI: 10.1016/j.physd.2017.01.003
 Authors: Rhys Bury; Alexander V. Mikhailov; Jing Ping Wang
 The stability spectrum for elliptic solutions to the focusing NLS equation
 Authors: Bernard Deconinck; Benjamin L. Segal
Abstract: Publication date: Available online 23 January 2017
Source:Physica D: Nonlinear Phenomena
Author(s): Bernard Deconinck, Benjamin L. Segal
We present an analysis of the stability spectrum of all stationary elliptictype solutions to the focusing Nonlinear Schrödinger equation (NLS). An analytical expression for the spectrum is given. From this expression, various quantitative and qualitative results about the spectrum are derived. Specifically, the solution parameter space is shown to be split into four regions of distinct qualitative behavior of the spectrum. Additional results on the stability of solutions with respect to perturbations of an integer multiple of the period are given, as well as a procedure for approximating the greatest real part of the spectrum.
PubDate: 20170128T05:15:28Z
DOI: 10.1016/j.physd.2017.01.004
 Authors: Bernard Deconinck; Benjamin L. Segal
 Reducedspace Gaussian Process Regression for datadriven probabilistic
forecast of chaotic dynamical systems Authors: Zhong Yi Wan; Themistoklis P. Sapsis
Abstract: Publication date: Available online 5 January 2017
Source:Physica D: Nonlinear Phenomena
Author(s): Zhong Yi Wan, Themistoklis P. Sapsis
We formulate a reducedorder strategy for efficiently forecasting complex highdimensional dynamical systems entirely based on data streams. The first step of our method involves reconstructing the dynamics in a reducedorder subspace of choice using Gaussian Process Regression (GPR). GPR simultaneously allows for reconstruction of the vector field and more importantly, estimation of local uncertainty. The latter is due to (i) local interpolation error and (ii) truncation of the highdimensional phase space. This uncertainty component can be analytically quantified in terms of the GPR hyperparameters. In the second step we formulate stochastic models that explicitly take into account the reconstructed dynamics and their uncertainty. For regions of the attractor which are not sufficiently sampled for our GPR framework to be effective, an adaptive blended scheme is formulated to enforce correct statistical steady state properties, matching those of the real data. We examine the effectiveness of the proposed method to complex systems including the Lorenz 96, the KuramotoSivashinsky, as well as a prototype climate model. We also study the performance of the proposed approach as the intrinsic dimensionality of the system attractor increases in highly turbulent regimes.
PubDate: 20170114T04:47:15Z
DOI: 10.1016/j.physd.2016.12.005
 Authors: Zhong Yi Wan; Themistoklis P. Sapsis
 Characterizing complex networks through statistics of Möbius
transformations Authors: Vladimir Jaćimović; Aladin Crnkić
Abstract: Publication date: Available online 31 December 2016
Source:Physica D: Nonlinear Phenomena
Author(s): Vladimir Jaćimović, Aladin Crnkić
It is wellknown now that dynamics of large populations of globally (alltoall) coupled oscillators can be reduced to lowdimensional submanifolds (WS transformation and OA ansatz). Marvel et al. (2009) described an intriguing algebraic structure standing behind this reduction: oscillators evolve by the action of the group of Möbius transformations. Of course, dynamics in complex networks of coupled oscillators is highly complex and not reducible. Still, closer look unveils that even in complex networks some (possibly overlapping) groups of oscillators evolve by Möbius transformations. In this paper we study properties of the network by identifying Möbius transformations in the dynamics of oscillators. This enables us to introduce some new (statistical) concepts that characterize the network. In particular, the notion of coherence of the network (or subnetwork) is proposed. This conceptual approach is meaningful for the broad class of networks, including those with timedelayed, noisy or mixed interactions. In this paper several simple (random) graphs are studied illustrating the meaning of the concepts introduced in the paper.
PubDate: 20170105T04:02:55Z
DOI: 10.1016/j.physd.2016.12.007
 Authors: Vladimir Jaćimović; Aladin Crnkić
 Classical quasisteady state reduction—A mathematical
characterization Authors: Alexandra Goeke; Sebastian Walcher; Eva Zerz
Abstract: Publication date: Available online 30 December 2016
Source:Physica D: Nonlinear Phenomena
Author(s): Alexandra Goeke, Sebastian Walcher, Eva Zerz
We discuss parameter dependent polynomial ordinary differential equations that model chemical reaction networks. By classical quasisteady state (QSS) reduction we understand the following familiar (heuristically motivated) mathematical procedure: Set the rate of change for certain (a priori chosen) variables equal to zero and use the resulting algebraic equations to obtain a system of smaller dimension for the remaining variables. This procedure will generally be valid only for certain parameter ranges. We start by showing that the reduction is accurate if and only if the corresponding parameter is what we call a QSS parameter value, and that the reduction is approximately accurate if and only if the corresponding parameter is close to a QSS parameter value. The QSS parameter values can be characterized by polynomial equations and inequations, hence parameter ranges for which QSS reduction is valid are accessible in an algorithmic manner. A defining characteristic of a QSS parameter value is that the algebraic variety defined by the QSS relations is invariant for the differential equation. A closer investigation of the associated systems shows the existence of further invariant sets; here singular perturbations enter the picture in a natural manner. We compare QSS reduction and singular perturbation reduction, and show that, while they do not agree in general, they do, up to lowest order in a small parameter, for a quite large and relevant class of examples. This observation, in turn, allows the computation of QSS reductions even in cases where an explicit resolution of the polynomial equations is not possible.
PubDate: 20170105T04:02:55Z
DOI: 10.1016/j.physd.2016.12.002
 Authors: Alexandra Goeke; Sebastian Walcher; Eva Zerz
 Cascades of alternating pitchfork and flip bifurcations in Hbridge
inverters Authors: Viktor Avrutin; Zhanybai T. Zhusubaliyev; Erik Mosekilde
Abstract: Publication date: Available online 28 December 2016
Source:Physica D: Nonlinear Phenomena
Author(s): Viktor Avrutin, Zhanybai T. Zhusubaliyev, Erik Mosekilde
Power electronic DC/AC converters (inverters) play an important role in modern power engineering. These systems are also of considerable theoretical interest because their dynamics is influenced by the presence of two vastly different forcing frequencies. As a consequence, inverter systems may be modeled in terms of piecewise smooth maps with an extremely high number of switching manifolds. We have recently shown that models of this type can demonstrate a complicated bifurcation structure associated with the occurrence of border collisions. Considering the example of a PWM Hbridge singlephase inverter, the present paper discusses a number of unusual phenomena that can occur in piecewise smooth maps with a very large number of switching manifolds. We show in particular how smooth (pitchfork and flip) bifurcations may form a macroscopic pattern that stretches across the overall bifurcation structure. We explain the observed bifurcation phenomena, show under which conditions they occur, and describe them quantitatively by means of an analytic approximation.
PubDate: 20170105T04:02:55Z
DOI: 10.1016/j.physd.2016.12.008
 Authors: Viktor Avrutin; Zhanybai T. Zhusubaliyev; Erik Mosekilde
 Spiral disk packings
 Authors: Yoshikazu Yamagishi; Takamichi Sushida
Abstract: Publication date: Available online 26 December 2016
Source:Physica D: Nonlinear Phenomena
Author(s): Yoshikazu Yamagishi, Takamichi Sushida
It is shown that van Iterson’s metric for disk packings, proposed in 1907 in the study of a centric model of spiral phyllotaxis, defines a bounded distance function in the plane. This metric is also related to the bifurcation of Voronoi tilings for logarithmic spiral lattices, through the continued fraction expansion of the divergence angle. The phase diagrams of disk packings and Voronoi tilings for logarithmic spirals are dual graphs to each other. This gives a rigorous proof that van Iterson’s diagram in the centric model is connected and simply connected. It is a nonlinear analog of the duality between the phase diagrams for disk packings and Voronoi tilings on the linear lattices, having the modular group symmetry.
PubDate: 20170105T04:02:55Z
DOI: 10.1016/j.physd.2016.12.003
 Authors: Yoshikazu Yamagishi; Takamichi Sushida